Multivalued dependency

According to database theory, a multivalued dependency is a full constraint between two sets of attributes in a relation.Let R {displaystyle R} be a relational schema and let α ⊆ R {displaystyle alpha subseteq R} and β ⊆ R {displaystyle eta subseteq R} (subsets). The multivalued dependency α ↠ β {displaystyle alpha woheadrightarrow eta } (which can be read as α {displaystyle alpha } multidetermines β {displaystyle eta } ) holds on R {displaystyle R} if, in any legal relation r ( R ) {displaystyle r(R)} , for all pairs of tuples t 1 {displaystyle t_{1}} and t 2 {displaystyle t_{2}} in r {displaystyle r} such that t 1 [ α ] = t 2 [ α ] {displaystyle t_{1}=t_{2}} , there exist tuples t 3 {displaystyle t_{3}} and t 4 {displaystyle t_{4}} in r {displaystyle r} such that t 1 [ α ] = t 2 [ α ] = t 3 [ α ] = t 4 [ α ] {displaystyle t_{1}=t_{2}=t_{3}=t_{4}} t 3 [ β ] = t 1 [ β ] {displaystyle t_{3}=t_{1}} t 3 [ R − β ] = t 2 [ R − β ] {displaystyle t_{3}=t_{2}} t 4 [ β ] = t 2 [ β ] {displaystyle t_{4}=t_{2}} t 4 [ R − β ] = t 1 [ R − β ] {displaystyle t_{4}=t_{1}} According to database theory, a multivalued dependency is a full constraint between two sets of attributes in a relation. In contrast to the functional dependency, the multivalued dependency requires that certain tuples be present in a relation. Therefore, a multivalued dependency is a special case of tuple-generating dependency. The multivalued dependency plays a role in the 4NF database normalization. A multivalued dependency is a special case of a join dependency, with only two sets of values involved, i.e. it is a binary join dependency. A multivalued dependency exists when there are at least three attributes (like X,Y and Z) in a relation and for a value of X there is a well defined set of values of Y and a well defined set of values of Z. However, the set of values of Y is independent of set Z and vice versa. The formal definition is given as follows. In more simple words the above condition can be expressed as follows: if we denote by ( x , y , z ) {displaystyle (x,y,z)} the tuple having values for α , {displaystyle alpha ,} β , {displaystyle eta ,} R − α − β {displaystyle R-alpha -eta } collectively equal to x , {displaystyle x,} y , {displaystyle y,} z , {displaystyle z,} correspondingly, then whenever the tuples ( a , b , c ) {displaystyle (a,b,c)} and ( a , d , e ) {displaystyle (a,d,e)} exist in r {displaystyle r} , the tuples ( a , b , e ) {displaystyle (a,b,e)} and ( a , d , c ) {displaystyle (a,d,c)} should also exist in r {displaystyle r} .